def _axiom_specialization_map_to_schema_instantiation_map(
propositional_specialization_map: PropositionalSpecializationMap,
substitution_map: Mapping[str, Formula]) -> Mapping[str, Formula]:
"""Converts the given propositional-logic specialization map from a
propositional-logic axiom to its specialization, to an instantiation map
from the schema equivalent of that axiom to a predicate-logic formula whose
skeleton is that specialization.
Parameters:
propositional_specialization_map: mapping specifying how some
propositional-logic axiom `axiom` (which is not specified) from
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM` specializes into
some specialization `specialization` (which is also not specified),
and containing no constants or operators beyond ``'~'``, ``'->'``,
``'|'``, and ``'&'``.
substitution_map: mapping from each atomic propositional subformula of
`specialization` to a predicate-logic formula.
Returns:
An instantiation map for instantiating the schema equivalent of `axiom`
into the predicate-logic formula obtained from its propositional
skeleton `specialization` by the given substitution map.
Examples:
>>> _axiom_specialization_map_to_schema_instantiation_map(
... {'p': PropositionalFormula.parse('(z1->z2)'),
... 'q': PropositionalFormula.parse('~z1')},
... {'z1': Formula.parse('Ax[(x=5&M())]'),
... 'z2': Formula.parse('R(f(8,9))')})
{'P': (Ax[(x=5&M())]->R(f(8,9))), 'Q': ~Ax[(x=5&M())]}
"""
for variable in propositional_specialization_map:
assert is_propositional_variable(variable)
for operator in propositional_specialization_map[variable].operators():
assert is_unary(operator) or is_binary(operator)
for key in substitution_map:
assert is_propositional_variable(key)
relation_to_formula = dict()
for key, formula in propositional_specialization_map.items():
relation_key = key.upper()
real_formula = Formula.from_propositional_skeleton(
formula, substitution_map)
relation_to_formula[relation_key] = real_formula
return relation_to_formula
def axiom_specialization_map_to_schema_instantiation_map(
propositional_specialization_map: PropositionalSpecializationMap,
substitution_map: Mapping[str, Formula]) -> Mapping[str, Formula]:
"""Converts the given propositional-logic specialization map from a
propositional axiom to its specialization, to an instantiation map from
the schema equivalent of that axiom to a predicate-logic formula whose
skeleton is that specialization.
Parameters:
propositional_specialization_map: map specifying how some propositional
axiom `axiom` (which is not specified) from
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM` specializes into
some specialization `specialization` (which is also not specified).
substitution_map: map from each atomic propositional subformula of
`specialization` to a predicate-logic formula.
Returns:
An instantiation map for instantiating the schema equivalent of `axiom`
into the predicate-logic formula obtained from its propositional
skeleton `specialization` by the given substitution map.
Examples:
>>> axiom_specialization_map_to_schema_instantiation_map(
... {'p': PropositionalFormula.parse('(z1->z2)'),
... 'q': PropositionalFormula.parse('~z1')},
... {'z1': Formula.parse('Ax[(x=5&M())]'),
... 'z2': Formula.parse('R(f(8,9))')})
{'P': (Ax[(x=5&M())]->R(f(8,9))), 'Q': ~Ax[(x=5&M())]}
"""
for variable in propositional_specialization_map:
assert is_propositional_variable(variable)
for key in substitution_map:
assert is_propositional_variable(key)
# Task 9.11.1
new_spec = {}
for key, val in propositional_specialization_map.items():
new_key = str.upper(key[0]) + key[1:]
new_val = Formula.from_propositional_skeleton(val, substitution_map)
new_spec[new_key] = new_val
return new_spec